Binary Calculator

To use Binary Calculator, enter the values in the input boxes below and click on Calculate button.

About Binary Calculator

Dive into the intricacies of the Arbitrary-Precision Binary Calculator, a versatile tool designed to perform operations on binary numbers with precision. This calculator is engineered to handle extensive integers and fractional values, both large and minuscule, offering a comprehensive platform for binary calculations.

Understanding the Calculator's Design

The calculator is intentionally kept simple, focusing on "pure" binary numbers without delving into computer number formats like two's complement or IEEE binary floating-point. This design choice ensures a clear exploration of binary operations in their fundamental state.

Navigating the Calculator's Features

Input

1. Operand Entry: Input positive or negative numbers without commas or spaces, avoiding scientific notation. Fractional values are denoted with a radix point ('.'), and negative numbers are prefixed with a minus sign ("-").

2. Operation Selection: Choose from addition (+), subtraction (-), multiplication (*), or division (/).

3. Bit Display: Adjust the number of bits displayed in the binary result, applicable primarily to division with an infinite fractional part.

4. Calculate: Execute the operation with a click on 'Calculate.'

5. Clear: Reset the form entirely to start afresh.

Output

Besides presenting the result of the operation, the calculator displays the number of digits in the operands and the result. For example, when multiplying 1.1101 by 111.100011, resulting in 1101.1010110111, the "Num Digits" box provides insights like "1.4 * 3.6 = 4.10." This breakdown signifies the digit count in the integer and fractional parts of each operand and the result.

Dealing with Precision and Infinite Results

1. Finite Results: Addition, subtraction, and multiplication invariably yield finite results.

2. Infinite Results in Division: Division often produces an infinite (repeating) fractional value. The calculator truncates, not rounds, infinite results to the specified number of bits. An ellipsis (...) indicates an infinite result, and '∞' symbolizes the number of fractional digits.

Exploring Floating-Point Arithmetic

While the calculator operates on pure binary arithmetic, it serves as a valuable tool for delving into floating-point arithmetic. For instance, you can use it to dissect imprecisions introduced by IEEE double-precision binary floating-point arithmetic.

Discussion: A Practical Example

An insightful example involves understanding why, in IEEE double-precision binary floating-point arithmetic, 129.95 * 10 yields 1299.5, while 129.95 * 100 results in 12994.999999999998181010596454143524169921875. The discussion emphasizes the sources of imprecision, such as decimal to floating-point conversion and limited-precision binary arithmetic.

Embracing the Role of a Binary Computer

To comprehend these intricacies, users emulate a computer's computational steps, from converting operands to binary to performing calculations and rounding results. This process unveils the precision challenges inherent in binary arithmetic.

Recognizing Limits

Practical limitations exist regarding the size of inputs and the number of fractional bits in infinite division results. Users exceeding these limits receive error messages, ensuring accurate results within the calculator's operational boundaries.